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A070162 Numbers k such that k - phi(k) - 1 is a prime.

%I A070162 #17 Nov 07 2024 08:33:00
%S A070162 6,8,9,10,12,14,16,18,20,22,26,34,36,38,40,42,44,46,48,50,56,58,60,62,
%T A070162 64,72,74,78,80,82,84,86,88,92,94,100,106,108,116,118,122,126,134,136,
%U A070162 142,146,150,152,156,158,162,164,166,178,180,182,192,194,198,202,204
%N A070162 Numbers k such that k - phi(k) - 1 is a prime.
%C A070162 Maximal solution is either k = 2p or, in the case of Mersenne primes M, k = 2(M+1) and f(k) = p or f(k) = M.
%H A070162 Amiram Eldar, <a href="/A070162/b070162.txt">Table of n, a(n) for n = 1..10000</a>
%F A070162 f(k) = k - A000010(k) - 1 = A051953(k) - 1 is prime.
%e A070162 k = 192: phi(192) = 64, cototient(192) = 128, k - phi(192) - 1 = 127 is prime.
%e A070162 k = 2p: 2p - phi(2p) - 1 = 2p - p + 1 - 1 = p, so 2*prime is always a solution.
%e A070162 k = 2^(q+1), where q is a Mersenne prime exponent: cototient(k) - 1 = 2^(p+1) - 2^p - 1 = 2^p - 1, which is the corresponding Mersenne prime.
%e A070162 Numbers k in {192, 224, 248, 254, 256} give p = 127.
%e A070162 numbers k in {72, 80, 88, 92, 94} give p = 47.
%t A070162 Do[s=n-EulerPhi[n]-1; If[PrimeQ[s], Print[n, s]], n, 1, 10000]
%t A070162 Select[Range[250],PrimeQ[#-EulerPhi[#]-1]&] (* _Harvey P. Dale_, Sep 05 2023 *)
%o A070162 (PARI) is(k) = isprime(k - eulerphi(k) - 1); \\ _Amiram Eldar_, Nov 07 2024
%Y A070162 Cf. A000010, A051953.
%K A070162 nonn
%O A070162 1,1
%A A070162 _Labos Elemer_, Apr 26 2002